Is there an explicit formula for a cosine function of x-coordinate? A question on ideas in Spivak's Calculus chapter on trigonometric functions.

Question

Spivak's Calculus has a Chapter 15 entitled "Trigonometric Functions". He starts with a discussion which I found a bit convoluted. I will describe how I understood the flow of ideas and concepts and at the end I will pose two questions that I have.

The idea of an angle is as simple as the union of two half-lines with a common initial point.

A directed angle can be thought of as a pair of half lines with the same initial point

Okay, not sure what the difference is between the two.

Next, it seems Spivak tries to motivate what will ultimately be the definitions of sine and cosine with some initial ideas.

Idea 1

We can specify a directed angle as a single point on the unit circle, ie a point $(x,y)$ with $x^2+y^2=1$.

We can then define sine and cosine of a directed angle as the $y$ and $x$ coordinates of the point representing the directed angle.

If we specify the directed angle by a number (degrees or radians), we could define sine and cosine functions. Given the directed angle as a number, we know the point on the unit circle, given this point we know cosine and sine.

In particular, if we measure the directed angle using radians, we are effectively giving each directed angle a number equal to arc length.

Hence, for each arc length we get a point, and given a point we get cosine and sine.

Okay, so the above is sort of an idea of what would happen if we were to define cosine and sine as simple x and y coordinates of a point on the unit circle, where the point is determined by a directed angle that is measured in radians.

Idea 2

What if instead of specifying an arc length, we were to specify an area of a sector on the unit circle (by the way, of course, starting at position $(1,0)$ moving counter-clockwise).

Given an arc length $x$, the sector area is $\frac{x}{2}$. Thus, instead of saying

$\sin x$ and $\cos x$ are the coordinates of the point determined by arc length $x$, where $x$ is in radians

we could say

$\sin x$ and $\cos x$ are the coordinates of the point determined by a sector area of $\frac{x}{2}$, where $x$ is in radians

Okay, at this point, the motivational ideas are finished and the formal definitions begin. I assume that at this point the "definitions" above were informal, and right now we start again from scratch, with $\cos$ and $\sin$ not yet defined.

Formal Definitions

First off, $\pi$ is defined as

$$\pi=2\int_{-1}^1 \sqrt{1-x^2}dx$$

Then, an area function is defined.

$$A(x)=\frac{x\sqrt{1-x^2}}{2}+\int_x^1 \sqrt{1-t^2}dt$$

This is just the function giving the area of a sector of the unit circle given an x-coordinate:

This function has certain characteristics: it is continuous on $[-1,1]$, is decreasing and hence one-one.

So at this point $\cos x$ is defined as the unique number in $[-1,1]$ such that

$$A(\cos{x})=\frac{x}{2}\tag{1}$$ and $\sin{x}$ is defined as

$$\sin{x}=\sqrt{1-\cos^2{x}}$$

Then comes a theorem that shows that $\cos'{x}=-\sin{x}$ and $\sin'{x}=\cos{x}$.

I have two questions

1. Is there a formula for $\cos{x}$, or is it just defined implicitly as in $(1)$?

2. Could we have defined $\cos$ in the following way

• instead of an area function we define an arc length function that given an x-coordinate would give us the arc-length (would this be done with a line integral?)
• find the inverse of such a function; ie a function that given an arc length gives us the x-coordinate.
• define $\cos x$ as the latter function
• define $\sin x$ as before

Answer 1

1. Equation (1) is pretty explicit, it says $\cos x:=A^{-1}\left(\frac{x}{2}\right)$, and you know the inverse exists because $A$ is injective, so you can restrict the target space to make it bijective. If however you mean a function defined as a derivative/integral of a polynomial or rational function of some kind, then no.
2. There are several ways of defining the trig functions. At the end of the chapter (right before the problems), Spivak talks about how one could take the primitive notion to be that of the $\arcsin$ function, defined as the integral $\int_0^x\frac{dt}{\sqrt{1-t^2}}$, and then defining $\sin$ as the inverse (of course a-priori it's only defined on a small subset of $\Bbb{R}$). See also the exercises in Spivak. One of them goes through the arclength approach, and another goes by defining $\arctan$ first. To see some of the geometry of $\arctan$, see Is it possible to find the coordinates of a point on the circumference of a circle, without using trigonometric functions?

In any case, the theme in this chapter (and later in Chapter 18 on logarithms and exponentials) is to define a function by an integral, prove theorems about the integral, and invert it to construct new functions. Spivak does things this way because it's usually more geometrically obvious. Of course, you can also introduce things the super slick way via power series (but Spivak isn't there yet). Or, if you're willing to take the existence theorem for ODEs on faith, you can also define trig and exponential functions in that manner (also outlined in the problems). So summary: keep reading and solving the problems.